MAJOR P. A. M.\. MAHOX ON THE COMPOSITIONS OF NUMBERS. 

 Hence the forms F,,, include the forms N m _. each 



(n m + 8\ ,. 



1 times. 



Hence 



F M = N M +( H - +I )N W _ I+ ( W -: 



Thence it is easy to show that 



and also 



The relation SN W = n! may he verified. 



Art. 24. It follows at once, from the zig-zag graphs, that 



N. = N._ M+I . 

 Some of the simplest results are 



81 



Art. 25. There is another interesting series for N,,. 

 Let 



denote the expansion ot 



when deprived of the term which is linear in p and of the term independent of p ; 



and put 



P. = (l+*)"-,.o ; 

 then 



N -P - 



- 1 



1 V 

 ' 



1 prove a general theorem, of which this is a particular case, later on in the paper. 

 VOL. ccvii. A. M 



